Numbers without zero

explanation.md

Non-zero numbers

This is just something I was toying with—what would our number system be like without a zero digit (which, presumably, was the case before 0 was invented).

I don't think it means anything, but the standard algorithms for addition and multiplication still work.

nonzero.shard.rb

# Non-zero numbers.  Say, in binary.  So two symbols, 1 & 2.

bnz  = %w[ 1   2  11   12   21   22  111   112   121   122   211   212   221   222  1111   1112 ]
ints =   [ 1,  2,  3,   4,   5,   6,   7,    8,    9,   10,   11,   12,   13,   14,   15,    16 ]
bz   = %w[ 1  10  11  100  101  110  111  1000  1001  1010  1011  1100  1101  1110  1111  10000 ]

# Note that in binary, 1 = 01 = 001 = 0001, etc.  But in non-zero numbers, 1 != 21 != 221, etc.

# Powers of 2: 2 -> 12 -> 112 -> 1112
# Multiples of 2: 2 -> 12 -> 22 -> 112 -> 122 -> 212 -> 222 -> 1112
# Multiples of 3: 11 -> 22 -> 121 -> 212 -> 1111
# Multiples of 4: 

# x << 1 = double x + 1
# x << 2 = double x + 2

class Fixnum
  def to_snz(base = 2)
    return "0" if self == 0    
    bits = 0
    offset = 0
    
    until self >= offset && self < offset + 2 ** bits
      offset += 2 ** bits
      bits   += 1
    end
    
    delta = self - offset
    deltas = delta.to_s(2)
    deltas = deltas.rjust(bits, '0')
    deltas = deltas.tr '01', '12'
  end
end

100.times do |i|
  # puts "#{i}\t#{i.to_snz(2)}\t#{i.to_s(3)}"
end

# Verify:
# ints.each_with_index do |int, idx|
#   binary          = int.to_s(2)
#   binary_non_zero = int.to_snz(2)
#   
#   puts "i = #{ int }\t#{ binary_non_zero == bnz[idx] ? "OK" : "FAIL" }"
# end

20.times do |i|
  m2 = (2*i).to_snz
  m3 = (3*i).to_snz
  m4 = (4*i).to_snz
  m5 = (5*i).to_snz
  
  puts "#{ i }\t#{i.to_snz}\t#{m2}\t#{m3}\t#{m4}\t#{m5}"
end

puts "#{ 123.to_snz } + #{ 345.to_snz } = #{ (123+345).to_snz }"
# require 'prime'
# 10_000.times do |i|
#   # if i.prime?
#     puts "#{i}\t#{i.to_snz}\t#{i.to_s(2)}"
#   # end
# end